We show that for every intermediate \ s-degree there exists an incomparable \ s-degree. As a consequence, for every intermediate \ Q-degree there exists an incomparable \ Q-degree. We also show how these results can be applied to provide proofs or new proofs of upper density results in local structures of s-degrees and Q-degrees.

We show that the theory of the local structure of the enumeration degrees is computably isomorphic to the theory of first order arithmetic. We introduce a novel coding method, using the notion of a K-pair, to code a large class of countable relations.

We show that every splitting of ${0}_{\mathrm{e}}^{\prime }$ in the local structure of the enumeration degrees, $$\mathcal{G}_{e} , contains at least one low-cuppable member. We apply this new structural property to show that the classes of all $\mathcal{K}$ -pairs in $\mathcal{G}_{e}$ , all downwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees and all upwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees are first order definable in $\mathcal{G}_{e}$.

We prove that a sequence of sets containing representatives of cupping partners for every nonzero ${\Delta^0_2}$ enumeration degree cannot have a ${\Delta^0_2}$ enumeration. We also prove that no subclass of the ${\Sigma^0_2}$ enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to ${\mathbf{0}_e'}$ by a single incomplete ${\Sigma^0_2}$ enumeration degree.

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) (...) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category. (shrink)

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is (...) a locally o-minimal structure which admits local definable cell decomposition. (shrink)

We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core. We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core.

In a bipartite quantum system, quantum states are classified as classically correlated and quantum correlated states, the later are important resources of quantum information and computation protocols. Since correlations of quantum states may vary under a quantum channel, it is necessary to explore the influence of quantum channels on correlations of quantum states. In this paper, we discuss CC-preserving, QC-breaking and strongly CC-preserving local quantum channels of the form \ and obtain the structures of these three types of (...) class='Hi'>local quantum channels. Moreover, we obtain a necessary and sufficient condition for a quantum state to be transformed into a CC state by a specific local channel \ in terms of the structure of the input quantum state. Lastly, as applications of the obtained results, we present a classification of local quantum channels \ and describe the quantum states which are transformed as CC ones by the corresponding local quantum channel. (shrink)

In substructural logics, structural principles may hold in some fragments of a consequence relation without holding globally. I look at this phenomenon in my preferred substructural logic, in which Weakening and Cut fail but which is supra-intuitionistic. I introduce object language operators that keep track of the admissibility of Weakening and of intuitionistic implications. I end with some ideas about local transitivity.

There is a long-standing and rarely contested view that Gricean conversational reasoning—the kind of reasoning that supports the identification of conversational implicatures—cannot produce pragmatically generated modification of the contents of embedded clauses. The goal of this paper is to argue against this view: to argue that embedded pragmatic effects can be seen as continuous with ordinary, utterance-level, conversational implicature. I will further suggest, though, that embedded pragmatic effects do force on us a particular conception of semantics. Specifically, I will argue (...) that an adequate account of the data requires a semantic framework that posits structured contents. (shrink)

We show that if M is a strongly minimal large homogeneous structure in a countable similarity type and the pregeometry of M is locally modular but not modular, then the pregeometry is affine over a division ring.

We examine recent work in cognitive neuroscience that investigates brain networks. Brain networks are characterized by the ways in which brain regions are functionally and anatomically connected to one another. Cognitive neuroscientists use various noninvasive techniques (e.g., fMRI) to investigate these networks. They represent them formally as graphs. And they use various graph theoretic techniques to analyze them further. We distinguish between knowledge of the graph theoretic structure of such networks (structural knowledge) and knowledge of what instantiates that (...) class='Hi'>structure (nonstructural knowledge). And we argue that this work provides structural knowledge of brain networks. We explore the significance of this conclusion for the scientific realism debate. We argue that our conclusion should not be understood as an instance of a global structural realist claim regarding the structure of the unobservable part of the world, but instead, as a local structural realist attitude towards brain networks in particular. And we argue that various local approaches to the realism debate, i.e., approaches that restrict realist commitments to particular theories and/or entities, are problematic insofar as they don't allow for the possibility of such a local structural realist attitude. (shrink)

Moti Mizrahi has argued that Thomas Kuhn does not have a good argument for the incommensurability of successive scientific paradigms. With Rouse, Andersen, and others, I defend a view on which Kuhn primarily was trying to explain scientific practice in Structure. Kuhn, like Hilary Putnam, incorporated sociological and psychological methods into his history of science. On Kuhn’s account, the education and initiation of scientists into a research tradition is a key element in scientific training and in his explanation of (...) incommensurability between research paradigms. The first part of this paper will explain and defend my reading of Kuhn. The second part will probe the extent to which Kuhn’s account can be supported, and the extent to which it rests on shaky premises. That investigation will center on Moti Mizrahi’s project, which aims to transform the Kuhnian account of science and of its history. While I do defend a modified kind of incommensurability, I agree that the strongest version of Kuhn’s account is steadfastly local and focused on the practice of science. (shrink)

The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.Institutionally, it was an honor to serve as President of the Association and I want to thank my teachers and predecessors for guidance and advice and my fellow officers and our publisher for their work and support. To all of the members who answered my calls to chair or serve on this or that committee, I offer my thanks as well. Your (...) work was both needed and appreciated.A major component of the efforts of the Association is devoted to our publications. My first important task as President was to deal with the need to reorganize the reviews section of the JSL and eventually to move it to the BSL. Appropriately enough, my first administrative job for the Association, some thirty years ago, was to serve on a committee to plan a reorganization of the reviews. I thank all those who helped with this transition and who took over the task of running the new reviews section. I hope that it will be another thirty years before further major changes are needed in this area and that someone else will be making them. (shrink)

There are $\Pi_5$ formulas in the language of the Turing degrees, D, with ≤, ∨ and $\vedge$ , that define the relations $x" \leq y"$ , x" = y" and so $x \in L_{2}(y)=\{x\geqy|x"=y"\}$ in any jump ideal containing $0^(\omega)$ . There are also $\Sigma_6$ & $\Pi_6$ and $\Pi_8$ formulas that define the relations w = x" and w = x', respectively, in any such ideal I. In the language with just ≤ the quantifier complexity of each of these definitions (...) increases by one. On the other hand, no Π2 or Σ2 formula in the language with just ≤ defines L2 or $x\inL_{2}(y)$ . Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of I is fixed on every degree above 0" and every relation on I that is invariant under double jump or joining with 0" is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in I. Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in I. Analogous results hold for various coarser degree structures. (shrink)

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o‐minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good‐shaped definable subsets called quasi‐special submanifolds. Using this fact, we investigate definably complete locally o‐minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o‐minimal expansions of (...) ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo‐definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo‐definable spaces are definably compact. Here, a pseudo‐definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set‐valued functions with definably compact domains of definition. (shrink)

For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of L1(G). We relate the notion (...) of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and we characterize property (T) for G in terms of the definability of the sets of fixed points of L1 functions on G. (shrink)

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\boldsymbol {\Pi }^{0}_{\alpha + 1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha $, $\alpha \geq 2$. We also study (...) approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures. (shrink)

The paper has two aims: (1) it sets out to show that it is well motivated to seek for an account of quantum non-locality in the framework of ontic structural realism (OSR), which integrates the notions of holism and non-separability that have been employed since the 1980s to achieve such an account. However, recent research shows that OSR on its own cannot provide such an account. Against this background, the paper argues that by applying OSR to the primitive ontology theories (...) of quantum physics, one can accomplish that task. In particular, Bohmian mechanics offers the best prospect for doing so. (2) In general, the paper seeks to bring OSR and the primitive ontology theories of quantum physics together: on the one hand, in order to be applicable to quantum mechanics, OSR has to consider what the quantum ontology of matter distributed in space-time is. On the other hand, as regards the primitive ontology theories, OSR provides the conceptual tools for these theories to answer the question of what the ontological status of the wave-function is. (shrink)

Research on human causal learning has largely focused on strength learning, or on computational-level theories; there are few formal algorithmic models of how people learn causal structure from covariations. We introduce a model that learns causal structure in a local manner via prediction-error learning. This local learning is then integrated dynamically into a unified representation of causal structure. The model uses computationally plausible approximations of rational learning, and so represents a hybrid between the associationist and (...) rational paradigms in causal learning research. We conclude by showing that the model provides a good fit to data from a previous experiment. (shrink)

We define a discrete closure operator for definably complete locally o‐minimal structures. The pair of the underlying set of and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it ‐dimension. A definable set X is of dimension equal to the ‐dimension of X. The structure is simultaneously a first‐order topological structure. The dimension rank of a set definable in the first‐order topological (...) structure also coincides with its dimension. (shrink)

We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.

We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.